Position Vectors: Finding Point Q on AB in 2:1 Ratio

[geoff18]

Homework Statement

Let A,B,C be the three points in R^3 with position vectors
a=(1,-2,0)
b=(-1,1,2)
c=(7,1,6)

respectively.

Find the position vector q of the point Q that divides the line segment AB in the ratio 2:1, (where Q is closer to B.)

Homework Equations

The Attempt at a Solution

i have no idea how to start...

 

[geoff18]

do i find the length of AB first and work out the ratio?

 

[HallsofIvy]

Homework Statement

Let A,B,C be the three points in R^3 with position vectors
a=(1,-2,0)
b=(-1,1,2)
c=(7,1,6)

respectively.

Find the position vector q of the point Q that divides the line segment AB in the ratio 2:1, (where Q is closer to B.)

Homework Equations

The Attempt at a Solution

i have no idea how to start...

do i find the length of AB first and work out the ratio?

No, it is not necessary to find the length of AB! You can, for instead, draw horizontal and vertical lines making a right triangle having vertices at A and B. Then "similar triangles" will make your job easy. Just divide the horizontal and vertal lines into 3 parts.

 

[geoff18]

i tried to do that, but i have no idea how to find the position vector of q...
some more hints pleasE? :p

 

[geoff18]

im waiting online for help, so any help is much appreciated.. thanks in advance. =)

 

[Mark44]

Write the vector equation for the line L that contains the segment AB. This equation will be r(t) = a + t*v

In this equation, a is the vector from the origin to point A, and v is the vector from point A to point B. t is the parameter, and r is a vector that goes from the origin to the point on the line L determined by the parameter t.

What vector is represented by r(0)? By r(1)? Can you think of a way to get to a point 2/3 the way along the segment AB?

 

[geoff18]

when r intersects the line?

 

[Mark44]

Each point of the line corresponds to r(t) for some value of t.

 

[geoff18]

so do i have to find t?
how do i find t?

 

[Mark44]

You get to pick t. If I choose t = 0, r(0) = a + 0*v = <1, -2, 0>. This vector goes from the origin to point A. What I'm calling v is the vector from A to B. Since I am multiplying by 0, I don't need to do any calculations with v for this value of t.

What is r(1)?

 

[Susanne217]

Homework Statement

Let A,B,C be the three points in R^3 with position vectors
a=(1,-2,0)
b=(-1,1,2)
c=(7,1,6)

respectively.

Find the position vector q of the point Q that divides the line segment AB in the ratio 2:1, (where Q is closer to B.)

Homework Equations

The Attempt at a Solution

i have no idea how to start...

Best way to show this is to show there exists a socalled linear combination from Linear Algebra of all three vectors which satisfies that condition.

if the vector are called $$v_1,v_2,v_3$$

then a linear combination is $$u_1 \cdot v_1 + u_2 \cdot v_2 + u_3 \cdot v_3$$

where the $$u_1,u_2,u_3$$ are weights...

 

[HallsofIvy]

Homework Statement

Let A,B,C be the three points in R^3 with position vectors
a=(1,-2,0)
b=(-1,1,2)
c=(7,1,6)

respectively.

Find the position vector q of the point Q that divides the line segment AB in the ratio 2:1, (where Q is closer to B.)

Homework Equations

The Attempt at a Solution

i have no idea how to start...

No, it is not necessary to find the length of AB! You can, for instead, draw horizontal and vertical lines making a right triangle having vertices at A and B. Then "similar triangles" will make your job easy. Just divide the horizontal and vertal lines into 3 parts.

Since you apparently did not understand my first response, using "similar triangles" based on the coordinate axes:
The change in x-coordinate form a(1, -2, 0) to b(-1, 1, 2) is -1- 1= -2. 2/3 of that is -4/3. That is, the change in x coordinate from a(1, -2, 0) to point Q is -4/3: the x coordinate of Q is 1+ -4/3= -1/3.

Do the same for the y and z coordinates.